Due to the ubiquity of Wi-Fi access points in buildings and urban areas in general, and the difficulty in obtaining reliable GPS signals in such areas, it is becoming increasingly common for portable devices to be able to determine their position with respect to a set of Wi-Fi access points of known position. For example, many smartphones are provided with access to a database that includes the identifiers and locations of Wi-Fi access points around the globe such that the smartphone can, on detecting one or more Wi-Fi access points, identify the locations of at least one of those access points (APs) and infer its location.
Typically, a portable Wi-Fi device may determine its position r by means of a simple weighted centroid that expresses the position of the device as a linear combination of the known locations ri of a set of Wi-Fi access points, each weighted by a weight wi that depends on the signal strength received from the respective access point scaled by its transmit power. This is illustrated in the below equation:
                    r        =                                            ∑              i                        ⁢                                                  ⁢                                          w                i                            ⁢                              r                i                                                                        ∑              i                        ⁢                                                  ⁢                          w              i                                                  1.1      
However, there can be considerable uncertainty in the locations of the Wi-Fi access points that conventional location determination algorithms do not take account of. This can lead to poor accuracy in the determined position of the device and an unknown uncertainty in the determined position of the device. Furthermore, the uncertainty in the location of an access point can have a significant directionality such that the location uncertainty is greater in one direction than in a perpendicular direction.
The same weighted centroid method of position determination can also be used to estimate the position of a Wi-Fi access point from the known locations of a set of Wi-Fi receivers receiving signals from the access point—or, equivalently, from multiple known locations of a single Wi-Fi receiver. For example, the positions of Wi-Fi access points are often determined through the use of a scanning vehicle fitted with a GPS receiver and a Wi-Fi receiver: as the scanning vehicle travels along a road it listens for Wi-Fi access points and infers the position of each Wi-Fi access point by measuring at multiple locations (as determined by its GPS receiver) the signal strength of signals from that access point. The position r of an access point can be estimated according to equation 1.1, where ri are the locations at which the Wi-Fi receiver makes a measurement of signal strength and wi is a weight that depends on the measured signal strength received from the respective access point.
As a result of the uncertainty in the GPS position of the Wi-Fi receiver (e.g. the scanning vehicle) and the uncertainties introduced by the method of position estimation, there can be considerable uncertainty in the estimated position of each access point. Furthermore, due to the fact that a Wi-Fi receiver may measure the signal strength of an access point from a limited number of locations, the uncertainty in the location of an access point can have a significant directionality—for example, a′ scanning vehicle might be constrained to move along a road such that the positional error of the access point in a direction perpendicular to the road is much greater than the error in a direction parallel to the road.
There is therefore a need for an improved method of forming an estimate of the position of a radio receiver relative to a set of radio transmitters whose locations have an associated uncertainty, and a need for an improved method of forming an estimate of the position of a radio transmitter relative to a set of radio receivers of known locations or relative to a single receiver present at multiple locations.